×

Über die Klassifikation numerischer Halbgruppen. (On the classification of numerical semigroups). (German) Zbl 0618.14008

Regensburger Mathematische Schriften 11. Regensburg: Univ. Regensburg, Fakultät für Mathematik. 81 p. (1987).
This paper studies the numerical semigroups, i.e. subsets H of \({\mathbb{N}}\) such that \(0\in H\), \(H+H\subseteq H\) and \({\mathbb{N}}-H\) is finite. It is well known that the study of numerical semigroups is closely related to the study of monomial curve singularities. In fact, for every numerical semigroup H one can associate the semigroup K-algebra \(K[[H]]=\{\sum_{h\in H}a_ ht^ h| \quad a_ h\in K\}\) (which is a K-subalgebra of K[[t]], where K is a fixed field). Many numerical invariants of the singularity K[[H]] such as the multiplicity, the embedding-dimension, the Hilbert function, etc. can be expressed in the language of semigroups. This point of view has been intensively studied by several authors e.g. Brezinski, Delorme, Herzog, Waldi, etc.
The author considers the set \(G_ m\) of numerical semigroups which contain the fixed number \(m\geq 3\), and associates to \(G_ m\) a convex polyhedral cone \(P_ m\subset {\mathbb{R}}^{m-1}\) in such a way that there is a bijection between \(G_ m\) and the set of points of \(P_ m\) with integral coordinates. This method has the advantage of translating the problem of classification of the semigroups of \(G_ m\) into a purely geometric language. In particular, one defines and studies the numerical invariant \(s_ m(H)\) which is defined as the dimension of the open face of \(P_ m\) containing the corresponding integral point of \(P_ m\) associated to H. This invariant is important because it gives a hierarchy of the semigroups of \(G_ m\). One also studies the Hilbert function of the integral points of a fixed face of \(P_ m.\)
In an appendix one gives the complete lists concerning the classification of the semigroups of \(G_ m\) with \(m\leq 7\) (by using the invariants studied before and a computer program).
Reviewer: L.Bădescu

MSC:

14H20 Singularities of curves, local rings
13H15 Multiplicity theory and related topics
20M10 General structure theory for semigroups
14-04 Software, source code, etc. for problems pertaining to algebraic geometry
PDF BibTeX XML Cite